Why is the mean used to compute the expectation in the GAN loss?
It seems your question is concerned with how an empirical mean works.
It is indeed true that, if all $x^{(i)}$ are independent identically distributed realisations of a random variable $X$, then $\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{i=1}^n f(x^{(i)}) = \mathbb{E}[f(X)]$. This is a standard result in statistics known as the law of large numbers.